In this section we perform the matching of this renormalizable, anomaly-free extension of the SM [1]} onto the effective theory SCET\(_{\rm BSM}\) . Assuming that the \(Z^{\prime }\)  boson will be the first new particle beyond the SM that is discovered experimentally, our setup provides a consistent framework for describing the two-body decays of the \(Z^{\prime }\)  into SM particles in a model-independent and systematic way. We describe the matching procedure in detail for the Wilson coefficients \(C_{\phi \phi }\)  and \(C_{\psi \psi }^{ij}\)  appearing at leading order in the effective Lagrangian (REF ). The fermionic operators are generated at tree level, while \(O_{\phi \phi }\)  occurs first at one-loop order (or via the kinetic mixing term). We first consider the case where the masses of the VLQs are of similar order as the mass of the \(Z^{\prime }\)  boson. In Section REF  we will then discuss the case of a double hierarchy \(m_X\gg m_{Z^{\prime }}\gg v\) .
